Chapter 6 Bessel functions. Bessel functions appear in a wide variety of physical problems. For example, separation of the Helmholtz or wave equation in circular cylindrical coordinates leads to Bessel’s equation. . Bessel’s eq. The solutions of the Bessel’s eq. are called Bessel functions.
Bessel functions appear in a wide variety of physical problems. For example, separation of the Helmholtz or wave equation in circular cylindrical coordinates leads to Bessel’s equation.
The solutions of the Bessel’s eq. are called Bessel functions.
In Chapter 3, we get the series solution of the above eq.
* Generating function, integer order, Jn (x)
Although Bessel functions are of interest primarily as solutions of differential equations, it is instructive and convenient to develop them from a completely different approach, that of the generating function. Let us introduce a function of two variables,
The coefficient of t n, Jn, is defined to be Bessel function of the first kind of integer order n. Expanding the exponentials, we have
Setting n =r- s, yields
for a positive integer m)
The coefficient of t n is then
This series form exhibits the behavior of the Bessel function Jn for small x. The results for J0, J1, and J2are shown in Fig.6.1. The Bessel functions oscillate but are not periodic.
Figure 6.1 Bessel function,
Since the terms for s<n (corresponding to the negative integer (s-n) ) vanish, the series can be considered to start with s=n. Replacing s by s + n, we obtain
These series expressions may be used with n replaced by v to define Jv and J-v for non-integer v.
* Recurrence relations
Differentiating Eq.(6.1) partially with respect to t, we findthat
and substituting Eq.(6.2) for the exponential and equating the coefficients of tn-1, we obtain
This is a three-term recurrence relation.
On the other hand, differentiating Eq.(6.1) partially with respect to x, we have
Again, substituting in Eq.(6.2) and equating the coefficients of t n , we obtain the result
As a special case,
Multiplying by xn and rearranging terms produces
Subtracting Eq.(6.12) from (6.10) and dividing by 2 yields
Multiplying by x-n and rearranging terms, we obtain
Please verify the follow result in class.
In particular, we have shown that the functions Jn defined by our generating functions, satisfy Bessel’s eq., and thus are indeed Bessel functions
A particular useful and powerful way of treating Bessel functions employs integral representations. If we return to the generating function (Eq. (6.2)), and substitute t=eiθ,
in which we have used the relations
and so on.
equating real and imaginary parts, respectively. It might be noted that angleθ (in radius) has no dimensions. Likewise sinθ has no dimensions and function cos(xsinθ) is perfectly proper from a dimensional point of view.
By employing the orthogonality properties of cousine and sine,
in which n and m are positive integers (zero is excluded), weobtain
If these two equations are added together
As a special case,
repeats itself in all four quadrants (
), we may write Eq. (6.30) as
On the other hand,
reverses its sign in the third and fourth quadrants
Adding Eq. (6.30a) and i times Eq. (6.30b), we obtain the complex exponential representation
This integral representation (Eq. (6.30c)) may be obtained somewhat more directly by employing contour integration.
In the theory of diffraction through a circular aperture we encounter the integral
, the amplitude of the diffracted wave. Here
is an azimuth angle in the
plane of the circular aperture of radius a,and
, is the angle defined by a
point on ascreen below the circular aperture relative to the normal through the center point. The parameter b is given by
the wavelength of the incident wave. The other symbols are
Fig.6.2 From Eq. (6.30c), we get
Equation (6.15) enables us to integrate Eq. (6.33) immediately toobtain
The intensity of the light in the diffraction pattern is proportional to
For v> 0, Jv (0)=0. Thus, for a finite interval [0, a], when
zero of Jv(i.e.
), we are able to have
This gives us orthogonality over the interval [0, a].
The normalization result may be written as
* Bessel series
If we assume that the set of Bessel functions
(v fixed, m=1,2,… )
may be expanded in a
is complete, then any well-behaved function
The coefficients cvm are determined by using Eq.(6.50),
* Continuum form
If a→∞, then the series forms may be expected to go over into integrals. The discrete roots become a continuous variable . A key relation is the Bessel function closure equation
From the theory of the differential equations it is known that Bessel’s equation has two independent solutions, Indeed, for non-integral order v we have already found two solutions and labeled themand ,using
theinfinite series (Eq. (6.5)). The trouble is that when v is integral Eq.(6.8) holds and we have but one independent solution. A second solution may be developed by the method of Section 3.6. This yields a perfectly good solution of Bessel’s equation but is not the usual standard form.
Definition and series form
As an alternate approach, we have the particular linear combination of \'
Bessel’s equation, for it is a linear combination of known solutions,
, our Neumann function or Bessel function of the second
To verify that
kind, actually does satisfy Bessel’s equation for integral n , we mayprocess
as follows. L’Hospital’s rule applied to Eq. (6.60) yields
with repect to v , we have
Multiplying the equation for
by(-1)v , substracting from the equation
(as suggested by Eq. (6.65)), and taking the limit
, we obtain
, an integer, the right-hand side vanishes by Eq. (6.8) and
is seen to be a solution of Bessel’s equation. The most general solution for
any v can be written as
Example Coaxial Wave Guides
We are interested in an electromagnetic wave confined between the conducting cylindrical surfaces
. Most of the mathematics is
inSection 3.3. From EM knowledge,
Let, we have
This is the Bessel equation. If ,the solution is
. But, for the coaxial wave guide one generalization
is needed. Theorigin is now excluded (). Hence the
Neumann functionmay not be excluded.
With the condition
we have the basic equatios for a TM (transverse magnetic ) wave.
The (tangential) electric field must vanish at the conducting surfaces(Direchlet
boundary condition) or
these transcendental equations may be solved for
and the ratio .
From the relation
and since must be positive for a real wave, the minimum
frequency that will
be propagated (in this TM mode) is
(6.82). This is
with fixed by the boundary conditions, Eqs. (6.81) and
the cutoff frequency of the wave guide.
Many authors perfer to introduce the Hankel functions by means of integral representations and then use them to define the Neumann function, .
We here introduce them a simple way as follows.
As we have already obtained the Neumann function by more elementary (and less powerful) techniques, we may use it to define the Hankel functions,
This is exactly analogous to taking
For real arguments and are complex conjugates. The extent of the analogy will be seen better when the asymptotic forms are considered . Indeed, it is their asymptotic behavior that makes the Hankel functuions useful!
6.5 Modified Bessel function , and
The Helmholtz equation,
separated in circular cylindrical coordinates, leads to Eq. (6.22a), the Bessel
equation. Equation (6.22a) is satisfied by the Bessel and Neumann functions
and any linear combination such as the Hankelfunctions
.Now the Helmholtz equation describes the space
part of wave phenomena. If instesd we have a diffusion problem, then the Helmholtz equation is replaced by
The Helmholtz equation may be transformed into the diffusion equation by the transformation . Similarly,changes Eq. (6.22a) into Eq. (6.89) and shows that
The solution of Eq. (6.89) are Bessel function of imaginary argument. To obtain a solution that is regular at the origin, we takeas the regular Bessel function
. It is customary (and convenient) to choose the normalization sothat
(Here the variable is being replaced by x for simplicity.) Often this is written as
In the terms of infinite series this is equivalent to removing the
sign in Eq. (6.5) and writing
normalization cancels the
from each term and leaves
real. For integral v this yields
The recurrence relations satisfied by
may be developed from the series
expansions,but it is easier to work from the existing recurrence relations for
. Let us replace x by –ix and rewrite Eq. (6.90) as
Repalcing x by ix, we have a recurrence relation for ,
Equation (6.12) transforms to
From Eq. (6.93) it is seen that we have but one independent solution when v is an integer, exactly as in the Bessel function solution of Eq. (6.108) is essentially a matter od convenience.
We choose to define a second solution in terms of the Hankel function
The factor makesreal when x is real. Using Eqs. (6.60) and (6.90), we may transform Eq. (6.97) to
analogous to Eq. (6.60) for The choice of Eq. (6.97) as a definition is somewhat unfortunate in that the function does not satisfy the same recurrence relations as . To avoid this annoyance other authors have included an additional factor of cos. This permits satisfy the same recurrence relations as , but it has the disadvantage of making for
To put the modified Bessel functions and in proper perspective, we introduce them here because:
1. These functions are solutions of the frequently encountered modified Bessel equation.
2. They are needed for specific physical problems such as diffusion problems.
6.6 Asmptotic behaviors
Frequently in physical problems there is a need to know how a given Bessel or modified Bessel functions for large values of argument, that is, the asymptotic behavior. Using the method of stepest descent studied in Chapter 2, we are able to derive the asymptotic behaviors of Hankel functions (see page 450 in the text book for details) and related functions:
2. The second kind Hankel function is just the complex conjugate of the
first (for real argument),
3. Since is the real part of
4. The Neumann function is the imaginary partof
5. Finally, the regular hyperbolic or modified Bessel function
is given by